Drorbn - User contributions [en]

08-401/Term Test

2008-04-05T01:40:07Z

Tout:

{{08-401/Navigation}}
==The Test==
===Front Page===
<center>

'''Do not turn this page until instructed.'''

<font style="font-size:150%">Math 401 Polynomial Equations and Fields</font>

<font style="font-size:125%">Term Test</font>

University of Toronto, February 27, 2008

'''Solve 5 of the 6 problems on the other side of this page.'''

Each of the problems is worth 20 points.

You have two hours to write this test.

'''Notes.'''

</center>

* No outside material other than stationary and a basic calculator is allowed.
* The final exam date was posted by the faculty - it will take place on the ''evening'' of Tuesday April 28 between 7PM and 10PM, at BN2S (Large Gymnasium, South End, Benson Building, 320 Huron Street (south of Harbord Street), Second Floor).
* '''Neatness counts! Language counts!''' The ''\em ideal'' written solution to a problem looks like a proof from the textbook; neat and clean and made of complete and grammatical sentences. Definitely phrases like "there exists" or "for every" cannot be skipped. Lectures are mostly made of spoken words, and so the blackboard part of proofs given during lectures often omits or shortens key phrases. The ideal written solution to a problem does not do that.
* '''New!''' You will get 20% of the credit for any problem for which you will write explicitly "I don't know how to solve this problem" (whole problems only!).

<center> '''Good Luck!''' </center>

===Questions Page===

'''Solve 5 of the following 6 problems.''' Each of the problems is worth 20 points. You have two hours. '''Neatness counts! Language counts!'''

'''Problem 1.''' Consider the ring <math>\{0,2,4,6,8\}</math> under addition and multiplication modulo 10. Does it have a unity?

<span style="font-size:80%">'''Tip.''' This, of course, is not just a yes/no question. You are expected to fully justify your answer, whatever it is.</span>

'''Problem 2.'''
# Define "an integral domain".
# Define "a field".
# Prove: A finite integral domain is a field.

<span style="font-size:80%">'''Tip.''' As always in math exams, when proving a theorem you may freely assume anything that preceded it but you may not assume anything that followed it.</span>

'''Problem 3.''' Let <math>R</math> be a commutative ring with unity and let <math>A</math> be an ideal in <math>R</math>. Prove that <math>R/A</math> is a field if and only if <math>A</math> is maximal.

<span style="font-size:80%">'''Tip.''' Don't forget! There are two directions to prove here!</span>

'''Problem 4.''' Find all ring homomorphisms from <math>{\mathbb Z}/6</math> to <math>{\mathbb Z}/10</math>.

<span style="font-size:80%">'''Tip.''' Here, of course, you have to explain both why the homomorphisms you found really are homomorphisms and why there are no more.</span>

'''Problem 5.'''
# Let <math>F</math> be a field, let <math>f</math> be a polynomial in <math>F[x]</math>, and let <math>a</math> and <math>b</math> be two different elements of <math>F</math>. Prove that the remainder for the division of <math>f</math> by <math>(x-a)(x-b)</math> is <math>\frac{f(b)-f(a)}{b-a}(x-a)+f(a)</math>.
# Compute the remainder for the division exercise <math>\frac{x^{2008}}{x^2-1}</math>, done in <math>{\mathbb Q}[x]</math>.

'''Problem 6.''' Prove that the polynomial <math>f=3x^5+15x^4-20x^3+10x+20</math> is irreducible over <math>{\mathbb Q}</math>. If you are using any major theorem, you need to quote it in full, but you don't need to prove it.

<center> '''Good Luck!''' </center>

{{08-401/Results of the Term Test}}

==Solution Set==

Solutions by Tout can be found [http://individual.utoronto.ca/tout/401TermTestSolution.pdf here]

Template:08-401/Navigation

2008-04-05T01:37:33Z

Tout:

{| cellpadding="0" cellspacing="0" style="clear: right; float: right"
|- align=right
|<div class="NavFrame"><div class="NavHead">[[08-401]]/[[Template:08-401/Navigation|Navigation Panel]]  </div>
<div class="NavContent">
{| border="1px" cellpadding="1" cellspacing="0" width="220" style="margin: 0 0 1em 0.5em; font-size: small"
|- align=center
!#
!Week of...
!Links
|- align=left
|align=center|1
|Jan 9
|[[08-401/About This Class|About]], [[08-401/Class Notes for January 9|Notes]], [[08-401/Homework Assignment 1|HW1]]
|- align=left
|align=center|2
|Jan 16
|[[08-401/Homework Assignment 2|HW2]], [[08-401/Class Notes for January 16|Notes]]
|- align=left
|align=center|3
|Jan 23
|[[08-401/Homework Assignment 3|HW3]], [[08-401/Class Photo|Photo]], [[08-401/Class Notes for January 23|Notes]]
|- align=left
|align=center|4
|Jan 30
|[[08-401/Homework Assignment 4|HW4]], [[08-401/Class Notes for January 30|Notes]]
|- align=left
|align=center|5
|Feb 6
|[[08-401/Homework Assignment 5|HW5]], [[08-401/Class Notes for February 6|Notes]]
|- align=left
|align=center|6
|Feb 13
|[[08-401/On the Term Test|On TT]], [[08-401/Class Notes for February 13|Notes]]
|- align=left
|align=center|R
|Feb 20
|Reading week
|- align=left
|align=center|7
|Feb 27
|[[08-401/Term Test|Term Test]], [http://individual.utoronto.ca/tout/401TermTestSolution.pdf Solutions]
|- align=left
|align=center|8
|Mar 5
|[[08-401/Homework Assignment 6|HW6]], [[08-401/Class Notes for March 5|Notes]]
|- align=left
|align=center|9
|Mar 12
|[[08-401/Homework Assignment 7|HW7]], [[08-401/Class Notes for March 12|Notes]]
|- align=left
|align=center|10
|Mar 19
|[[08-401/Homework Assignment 8|HW8]], [[08-401/Class Notes for March 19|Notes]], [[Media:08-401-RefCard.pdf|RC]] (PDF)
|- align=left
|align=center|11
|Mar 26
|[[08-401/Homework Assignment 9|HW9]], [[08-401/Class Notes for March 26|Notes]]
|- align=left
|align=center|12
|Apr 2
|[[08-401/The Fundamental Theorem|FT]], [[08-401/Homework Assignment 10 (and last!)|HW10]], [[08-401/Class Notes for April 2|Notes]]
|- align=left
|align=center|13
|Apr 9
|[[08-401/Class Notes for April 9|Notes]]
|- align=left
|align=center|S
|Apr 14-18
|Study Period
|- align=left
|align=center|F
|Apr 28
|[[08-401/The Final Exam|Final]]
|-
|colspan=3 align=center|[[Image:08-401 Class Photo.jpg|180px]]<br>[[08-401/Class Photo|Add your name / see who's in!]]
|-
|colspan=3 align=center|[[08-401/Register of Good Deeds|Register of Good Deeds]]
|}
</div></div>
|}
<div align=center style="color: red; font-size: 150%; display: none;">Announcements go here</div>

File:401TermTestSolution.pdf

2008-04-05T01:32:08Z

Tout:

08-401/Class Photo

2008-02-21T21:07:39Z

Tout:

{{08-401/Navigation}}

Our class on Janury 23, 2007:

[[Image:08-401 Class Photo.jpg|thumb|centre|500px|Class Photo: click to enlarge]]

Please identify yourself in this photo! There are two ways to do that:

* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.
* Send [[User:Drorbn|Dror]] an email message with this information.

The first option is more fun but less private.

If you are not in the photo, Dror can take your picture some other time and stitch you in.

===Who We Are...===

{| align=center border=1
|-
!First name
!Last name
!UserID
!Email
!In the photo
!Comments

{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn @ math.toronto.edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank}}
{{Photo Entry|last=Anwar|first=Farheen|userid=Farheen|email=farheen.anwar @ utoronto.ca|location= Only girl with head scarf, 4th from the right|comments=}}
{{Photo Entry|last=Bathelt|first=Jenny|userid=BatheltJ|email=jenny.bathelt @ utoronto.ca|location=Third from the left, front row, in the white sweater|comments=}}
{{Photo Entry|last=Cao|first=Ying|userid=McTea|email= y.cao[at]utoronto.ca|location= third from right, in front, blue scarf|comments= ^^}}
{{Photo Entry|last=Constantinescu|first=Sorin|userid=Sorin.c|email=first_name.last_name @ utoronto.ca|location=Sixth person from the right, blue striped dress shirt.|comments=If my shirt was red, I would look like Spiderman (sorta).}}
{{Photo Entry|last=Gershon|first=Amir|userid=LeoDaVinci|email=amir.gershon @ utoronto.ca|location='''Really''' tall bearded guy, back row, fourth from the right in the bright red shirt.|comments=<math>N=q^{\alpha} p_1^{2e_1} \ldots p_k^{2e_k}, </math><br><math>{\alpha} \equiv 1 \pmod 4</math>}}
{{Photo Entry|last=Ho|first=Shelley|userid=Mcgrill|email=shelley.ho @ utoronto.ca|location=the girl 5th from the right,front row|comments= =)}}
{{Photo Entry|last=Lan|first=Kemei|userid=Lankemei2002|email=lankemei2002 @ yahoo.com|location=2nd from the left with black short hair|comments= this is fun!}}
{{Photo Entry|last=Lee|first=Lori|userid=leelorik|email=leelorik @ gmail.com|location=6th from the left, in the front row|comments=2 + 2 = 4}}
{{Photo Entry|last=Leong|first=Alex|userid=AlexL|email=alex.leong @ utoronto.ca|location=middle, slightly to left, wearing plain white T|comments=}}
{{Photo Entry|last=Memon|first=Shaun|userid=s.memon|email=s.memon @ utoronto.ca|location=back row in the centre, I have 14 people to the left and 14 people to the right of me|comments=Shaun ∈ Z(MAT401)}}
{{Photo Entry|last=Metropolit|first=Geoff|userid=Geoffmetro|email=geoff.metropolit @ utoronto.ca|location=fourth from the left, back row|comments=}}
{{Photo Entry|last=Robertson|first=James|userid=Emailforjimmy|email=emailforjimmy NOSPAM @ gmail.com|location= I am the tall person (2nd only to the bearded fellow in red...Amir I think) near the middle in the back|comments= can you tell who just found out he handed in the wrong problems for Ass. # 1?}}
{{Photo Entry|last=Robinson|first=Patrick|userid=PRobinson|email=patrickjames.robinson at utoronto.ca|location= of the two chaps wearing red, I'm on the right with the waistcoat & glasses|comments=}}
{{Photo Entry|last=Sansern|first=Oak|userid=Hamlet|email=oak.sansern @ utoronto.ca|location=front row, 1st from the projector in white--looking too tired for eve class|comments=seriously? maybe later =)}}
{{Photo Entry|last=Selariu|first=Diana|userid=Diana|email=dselariu @ gmail.com|location=5th from the left, front row}}
{{Photo Entry|last=Shin|first=Tae Maen|userid=gopsury83|email= tim.shin @ utoronto.ca|location= second from right, back row|comments= You can call me Tim}}
{{Photo Entry|last=Skinner|first=Brian|userid=BrianSkinner|email= brian.skinner@ utoronto.ca|location= second from left, back row|comments= N/A}}
{{Photo Entry|last=Udasco|first=Chris|userid=Christopher|email=chris dot udasco at utoronto dot ca|location=front row grin|comments=2 + 2 = 1 (..huh?)<br><br><br>... Mod 3! (whaaaaat!!)}}
{{Photo Entry|last=Wang|first=Tout|userid=Tout|email=|location=back row, third from the left|comments=}}
{{Photo Entry|last=Yiu|first=Tony|userid=Tonytastic|email=tony.yiu @ utoronto.ca|location=Asian guy, left of the middle wearing blue|comments=Chuck Norris can divide by zero}}
{{Photo Entry|last=Yu|first=Jianhong|userid=yujian968|email=yujian968@ hotmail.com|location=A girl stands in the middle of the 1st row, with a black scarf|comments= ^..^! sinx/n=6}}